the cohomology of coherent sheaves - unirioja...mohamed barakat & markus lange-hegermann university...

OverviewAffine Schemes

Projective SchemesThe Standard Module

The Cohomology of Coherent Sheaves

Mohamed Barakat & Markus Lange-Hegermann

University of Kaiserslautern & RWTH-Aachen University

Mathematics Algorithms and Proofs 2010, Logroño

Mohamed Barakat & Markus Lange-Hegermann The Cohomology of Coherent Sheaves

http://homalg.math.rwth-aachen.de/index.php/extensions/gradedmoduleshttp://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://www.mathematik.uni-kl.dehttp://www.mathematik.rwth-aachen.dehttp://www.unirioja.es/dptos/dmc/MAP2010/http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves

1 Affine SchemesFrom rings to schemesFrom modules to quasi-coherent sheaves

2 Projective SchemesFrom graded rings to projective schemesFrom graded modules to quasi-coherent sheaves

3 The Standard ModuleThe module of global sections

http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves

From rings to schemesFrom modules to quasi-coherent sheaves

Overview

1 Affine SchemesFrom rings to schemes

From modules to quasi-coherent sheaves

2 Projective SchemesFrom graded rings to projective schemes

From graded modules to quasi-coherent sheaves

The spectrum of a ring

DefinitionLet R be a commutative ring with one. The set

Spec(R) := {p ⊳ R | p prime}

is called the (prime) spectrum of R. Define the vanishinglocus of an ideal I E R

V (I) := {p ∈ Spec(R) | p ⊃ I} = V (√I).

The set {V (√I) | I E R} forms the set of closed subsets of a

topology on X = Spec(R), the so-called ZARISKI topology :PROOF. V (R) = ∅, V (〈0〉) = Spec(R), V (I) ∪ V (J) = V (IJ),and

⋂V (Ii) = V (

∑Ii).

V (I) := {p ∈ Spec(R) | p ⊃ I} = V (√I).

⋂V (Ii) = V (

∑Ii).

V (I) := {p ∈ Spec(R) | p ⊃ I} = V (√I).

⋂V (Ii) = V (

∑Ii).

V (I) := {p ∈ Spec(R) | p ⊃ I} = V (√I).

⋂V (Ii) = V (

∑Ii).

The affine scheme associated to a ring

Definition

For the prime spectrum U := Spec(R) define the structuresheaf OU as the sheaf with stalks being the local ringsOU |p := Rp := (R \ p)−1R for all p ∈ U and ... . The sheaf ofrings OU is called the sheafification of the ring R. The locallyringed space (U,OU ) is called the affine scheme of R.

In ... I didn’t tell you how to define OU (U(I)) for a general openset U(I) := U \ V (I) (cf. [Har77, p. 70]). But for thedistinguished open sets of the form D(f) := U \ V (〈f〉) forf ∈ R, which form a basis of the ZARISKI topology:

OU (D(f)) := Rf = R[1f] = R[x]/〈xf − 1〉, OU (U) = R.

Definition

Example

The affine space Ank := Spec(k[x1, . . . , xn]).

For I E k[x1, . . . , xn] the affine subschemeV (I) := Spec(k[x1, . . . , xn]/I).

The two ingredients “prime spectrum” and “sheaf” preexistedthe definition of an affine scheme (U,OU ) but it wasGROTHENDIECK who put them together.

Corollary

{rings}op ≃ {affine schemes}.

Example

Corollary

Example

Corollary

Example

Corollary

Schemes

Definition

A scheme is a locally ringed space (X,OX ) in which each pointx ∈ X has an open neighborhood U such that the locally ringedspace (U,OX |U ) is an affine scheme.

We don’t know how to “glue” rings

{rings} ⊂ ? (algebra)

but we know how to glue topological spaces and how to gluetheir structure sheaves

{rings}op ≃ {affine schemes} ⊂ {schemes} (geometry)

Schemes

Definition

Schemes

Definition

The quasi-coherent sheaf associated to a module

For an R-module M define the sheafification M̃ to be thesheaf on Spec(R) satisfying

M̃p := Mp := (R \ p)−1M = Rp ⊗R M

and

M̃(D(f)) := Mf := Rf ⊗R M

for any p ∈ Spec(R) and any f ∈ R. M̃ is the prototype of aquasi-coherent sheaf.

Corollary

{R-modules} ≃˜

//{quasi-coherent sheaves on Spec(R)}

M̃p := Mp := (R \ p)−1M = Rp ⊗R M

and

M̃(D(f)) := Mf := Rf ⊗R M

Corollary

{R-modules} ≃˜

M̃p := Mp := (R \ p)−1M = Rp ⊗R M

and

M̃(D(f)) := Mf := Rf ⊗R M

Corollary

{R-modules} ≃˜

Quasi-coherent sheaves

Definition

Let (X,OX ) be a scheme. A sheaf of OX-modules F is calledquasi-coherent if X can be covered by open affine subsetsUi := Spec(Ri) with F |Ui∼= M̃i (where Mi is some Ri-module).

From graded rings to projective schemesFrom graded modules to quasi-coherent sheaves

Overview

The projective space Pnk := Proj(k[x0, . . . , xn])

We would like to construct Pnk as the scheme

Pnk “:=” (Ank \ {0})/k∗.

So consider the graded polynomial ring S := k[x0, . . . , xn]=

⊕i≥0 Si. Define the n+ 1 polynomial rings

Ri := k[x0xi, . . . , xn

xi] < k(Pnk) := k(

x1x0, . . . , xn

x0).

Define the projective space

Pnk := Proj(S)

by gluing together the affine spaces Ui := Spec(Ri) along theiraffine intersections Uij := Ui ∩ Uj := Spec(〈Ri, Rj〉), where〈Ri, Rj〉 is the subring of k(Pnk) generated by Ri and Rj.

Pnk “:=” (Ank \ {0})/k∗.

Ri := k[x0xi, . . . , xn

xi] < k(Pnk) := k(

x1x0, . . . , xn

x0).

Pnk := Proj(S)

Pnk “:=” (Ank \ {0})/k∗.

Ri := k[x0xi, . . . , xn

xi] < k(Pnk) := k(

x1x0, . . . , xn

x0).

Pnk := Proj(S)

Pnk “:=” (Ank \ {0})/k∗.

Ri := k[x0xi, . . . , xn

xi] < k(Pnk) := k(

x1x0, . . . , xn

x0).

Pnk := Proj(S)

The projective space P1k := Proj(k[x0, x1])

Example

Define the two polynomial rings R0 := k[x1x0 ] = k[t] andR1 := k[

x0x1] = k[t−1]. The projective line

P1k := Proj(k[x0, x1])

as the gluing of the affine spaces

U0 := Spec(R0) = Spec(k[t]) and

U1 := Spec(R1) = Spec(k[t−1])

along their affine intersection U01 := U0 ∩ U1 := Spec(〈R0, R1〉),where 〈R0, R1〉 = k[t, t−1] is LAURENT polynomial ring, which isthe subring of k(P1k) = k(

x1x0) = k(t) generated by Ri and Rj.

The projective space P1k := Proj(k[x0, x1])

Example

Define the two polynomial rings R0 := k[x1x0 ] = k[t] andR1 := k[

x0x1] = k[t−1]. The projective line

P1k := Proj(k[x0, x1])

as the gluing of the affine spaces

U0 := Spec(R0) = Spec(k[t]) and

U1 := Spec(R1) = Spec(k[t−1])

along their affine intersection U01 := U0 ∩ U1 := Spec(〈R0, R1〉),where 〈R0, R1〉 = k[t, t−1] is LAURENT polynomial ring, which isthe subring of k(P1k) = k(

x1x0) = k(t) generated by Ri and Rj.

The Proj construction

One can analogously define X := Proj(S) for an arbitrarygraded rings S =

⊕i≥0 Si. Let m := S>0 :=

⊕i>0 Si denote the

maximal graded ideal in S.

Define the set

Proj(S) := {p ⊳ S | p homogeneous prime and p 6⊃ m}.For a homogeneous ideal I E S set

V (I) = {p ∈ Proj(S) | p ⊃ I}.For p ∈ Proj(S) define the localization S(p) := (S \ p)−1homS.For a homogeneous f ∈ m define S(f) := (Sf )0 andD(f) := Proj(S) \ V (〈f〉) = Spec(S(f)).

Finally use the above to construct the structure sheaf OX .

⊕i≥0 Si. Let m := S>0 :=

Define the set

⊕i≥0 Si. Let m := S>0 :=

Define the set

⊕i≥0 Si. Let m := S>0 :=

Define the set

⊕i≥0 Si. Let m := S>0 :=

Define the set

⊕i≥0 Si. Let m := S>0 :=

Define the set

⊕i≥0 Si. Let m := S>0 :=

Define the set

⊕i≥0 Si. Let m := S>0 :=

Define the set

Projective schemes

Definition

A projective scheme X is a scheme of the form

X := Proj(S)

for some graded ring S =⊕

i≥0 Si.

The quasi-coh. sheaf associated to a graded module

For a graded S-module M =⊕

i∈ZMi define the sheafificationM̃ to be the quasi-coherent sheaf on Proj(S) satisfying

M̃p := M(p) :=((S \ p)−1homM

)0

and

M̃(D(f)) := M(f) := (Mf )0 := (Sf ⊗S M)0for any p ∈ Proj(S) and any homogeneous f ∈ m.

Theorem

Any quasi-coherent sheaf on a projective scheme X := Proj(S)is the sheafification M̃ of some graded S-module M .

The quasi-coh. sheaf associated to a graded module

For a graded S-module M =⊕

i∈ZMi define the sheafificationM̃ to be the quasi-coherent sheaf on Proj(S) satisfying

M̃p := M(p) :=((S \ p)−1homM

)0

and

M̃(D(f)) := M(f) := (Mf )0 := (Sf ⊗S M)0for any p ∈ Proj(S) and any homogeneous f ∈ m.

Theorem

Any quasi-coherent sheaf on a projective scheme X := Proj(S)is the sheafification M̃ of some graded S-module M .

Up to finite length

Unfortunately the sheafification does not yield an equivalenceof categories

{graded S-modules} 6≃˜

//{quasi-coh. sheaves onProj(S)}

Theorem

Two graded S-modules M and N define the samequasi-coherent sheaf iff M≥d ∼= N≥d for some d ∈ Z.

Q: Is there a way to find a canonical representative in anequivalence class of graded modules “isomorphic in highdegrees”?

Up to finite length

Theorem

Up to finite length

Theorem

The module of global sections

Overview

The global sections of the twisted sheaves

Let S be a graded NOETHERIAN ring and M a finitely generatedgraded S-module. Denote by M [i] the shifted S-module with

M [i]j := Mi+j

and by

H0(M̃ [i]) := M̃ [i](Proj(S))

the global sections of the coherent twisted sheaf M̃ [i].

M [i]j := Mi+j

and by

H0(M̃ [i]) := M̃ [i](Proj(S))

M [i]j := Mi+j

and by

H0(M̃ [i]) := M̃ [i](Proj(S))

The standard module

DefinitionThe truncated direct sum

H0≥0(M̃ ) :=⊕

i≥0H0(M̃ [i])

has a natural S-module structure. We call it the standardmodule of the coherent sheaf M̃ (truncated at 0).

Theorem

The sheafification of the standard module of a sheaf M̃ is againisomorphic to M̃ :

˜H0≥0(M̃ )

∼= M̃ .

The standard module

H0≥0(M̃ ) :=⊕

i≥0H0(M̃ [i])

Theorem

˜H0≥0(M̃ )

∼= M̃ .

The standard module

H0≥0(M̃ ) :=⊕

i≥0H0(M̃ [i])

Theorem

˜H0≥0(M̃ )

∼= M̃ .

software demo

Mohamed Barakat and Markus Lange-Hegermann, AnAxiomatic Setup for Algorithmic Homological Algebra andan Alternative Approach to Localization, to appear inJournal of Algebra and its Applications(arXiv:1003.1943).

Robin Hartshorne, Algebraic geometry, Springer-Verlag,New York, 1977, Graduate Texts in Mathematics, No. 52.MR MR0463157 (57 #3116)

http://arxiv.org/abs/1003.1943http://www.mathematik.uni-kl.de/~barakat/http://wwwb.math.rwth-aachen.de/markus/http://homalg.math.rwth-aachen.de/index.php/unreleased/sheaves

OverviewAffine SchemesFrom rings to schemesFrom modules to quasi-coherent sheaves

Projective SchemesFrom graded rings to projective schemesFrom graded modules to quasi-coherent sheaves

The Standard ModuleThe module of global sections

the cohomology of coherent sheaves - unirioja...mohamed barakat & markus lange-hegermann university...

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